Contents

supersymmetry

# Contents

## Idea

Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object $\mathbb{R}$ in there.

As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.

As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superschemes.

From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.

For more see at geometry of physics – supergeometry.

Supergeometry may defined over an arbitrary commutative ring (Schwarz and Shapiro 06).

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$$\phantom{A}$site$\phantom{A}$$\phantom{A}$sheaf topos$\phantom{A}$$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$$\phantom{A}$Point$\phantom{A}$$\phantom{A}$Set$\phantom{A}$$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$CartSp$\phantom{A}$$\phantom{A}$SmoothSet$\phantom{A}$$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$$\phantom{A}$FormalCartSp$\phantom{A}$$\phantom{A}$FormalSmoothSet$\phantom{A}$$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$SuperFormalCartSp$\phantom{A}$$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$$\phantom{A} \phantom{A}$
$\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

## References

### General

Historically influential general considerations:

Survey:

Introductory lecture notes:

Discussion via functorial geometry:

Lectures on supergeometry highlighting methods from algebraic geometry:

Discussion of traditional algebraic geometry for super-schemes:

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in

and in

• V. Molotkov, Infinite-dimensional $\mathbb{Z}_2^k$-supermanifolds, ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

• Anatoly Konechny, Albert Schwarz,

On $(k \oplus l|q)$-dimensional supermanifolds, in Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume), Lecture Notes in Physics 509, Springer (1998) [arXiv:hep-th/9706003, doi:10.1007/BFb0105247]

Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

Generalization to supergeometry over an arbitrary commutative ring, in particular $p$-adic rings, is given in

A review of all this as geometry in the topos over the category of superpoints is in

Formulation in terms of synthetic differential supergeometry is in

For many more references see at supermanifold.

Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in

especially in the contribution

The appendix there

• Sign manifesto (pdf)

means to sort out various sign conventions of relevance.

Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in

For more on this see at superalgebra.

Discussion related to G-structure and Killing spinors includes

Generalization of Artin's representability theorem to supergeometry:

For AQFT-like discussion of supersymmetric field theory:

### Supergeometry of fermion fields

Discussion of the classical mechanics of the spinning particle or of classical field theory with fermion fields (possibly but not necessarily super-symmetric) as taking place in supergeometry:

via (possibly infinite-dimensional) supermanifolds:

and more generally via smooth super sets:

Discussion with focus on supersymmetry:

and specifically in the context of super- string theory (regarding worldsheets as super Riemann surfaces):

Last revised on September 2, 2024 at 15:22:46. See the history of this page for a list of all contributions to it.